int B[maxn] , i = p , j = q+1 , k = 0;
while(k < r - p + 1) {
if(i > q || j <= r && A[i] > A[j]) B[k++] = A[j++];
else B[k++] = A[i++];
}
for(i=0;i<r-p+1;i++) A[p+i] = B[i];
}
2.3-5 浜屽垎鏌ユ壘鐨凜++浠ｇ爜
if(l == r) return l;
int mid = (l+r) >> 1;
if(a[mid] >= value) return find(a, l, mid, value);
else return find(a , mid+1, r , value);
}
*2.3-7 錛堣繖閬撻鍏跺疄鏈塐(n)鐨勭畻娉曪紝鑰屼笖鍐欒搗鏉ユ洿鏂逛究浜涳級榪欓噷鏄疧(nlogn)鐨勭畻娉?br /> O(nlogn)綆楁硶鎬濇兂錛?.棣栧厛榪涜鎺掑簭錛?.鐒跺悗鏋氫婦姣忎竴涓皬浜庣瓑浜巟/2鐨勬暟S[i]錛屼簩鍒嗘煡鎵懼搴旂殑x-S[i]鏄惁瀛樺湪
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;
const int maxn = 1010;
bool findx(int *S,int n, int x,int l,int r) {
if(l > r) return false;
if(l ==r) return S[l] == x;
int mid = (l+r) >> 1;
if(S[mid] >= x) return findx(S, n, x, l, mid);
else return findx(S, n, x, mid+1, r);
}
bool check(int *S,int n,int x) {
for(int i=0;S[i]<=x/2 && i < n;i++) {
if(findx(S, n, x-S[i], i+1, n-1)) return true;
}
return false;
}
int main() {
int S[1010] , x , n;
while(~scanf("%d%d" , &n , &x)) {
for(int i=0;i<n;i++) cin >> S[i];
if(check(S, n, x)) puts("yes");
else puts("no");
}
return 0;
}
O(n)鐨勬柟娉曟槸鍦ㄦ暟鐨勮寖鍥翠笉鏄壒鍒ぇ鐨勬椂鍊欙紙鎴栬呮暟鐨勮寖鍥存瘮杈冨ぇ錛屾鏃墮噰鐢╤ash鐨勬柟娉曪級鏍囪鐨勬柟娉曪紝榪欓噷鍋囪鏁扮殑鑼冨洿<=10000錛屽茍涓斿亣璁炬暟娌℃湁閲嶅鐨勬儏鍐典笅錛屽叾浠栨儏鍐電◢璁告敼鍙樹竴涓嬪氨琛岋細(xì)
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;
const int maxn = 1010;
bool check(int *S,int n,int x) {
bool vis[10001] = {0};
for(int i=0;i<n;i++) vis[x-S[i]] = true;
for(int i=0;i<n;i++) if(vis[S[i]]) return true;
return false;
}
int n ,x , S[maxn];
int main() {
while(~scanf("%d%d" , &n , &x)) {
for(int i=0;i<n;i++) cin >> S[i];
if(check(S, n, x)) puts("yes");
else puts("no");
}
return 0;
}
2-4錛堥嗗簭瀵癸級錛氳繖閬撻灝辨槸鍦ㄥ綊騫舵帓搴忎腑寰楀埌閫嗗簭瀵癸紝鍏蜂綋瑙佷唬鐮侊細(xì)
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;
const int maxn = 1010;
int ans;
void merge_sort(int *A, int l,int r) {
if(l >= r) return;
int mid = (l+r) >> 1;
merge_sort(A, l, mid);
merge_sort(A, mid+1, r);
int i = l , j = mid+1 ,B[maxn] , k = l;
while(i <= mid || j <= r) {
if(i > mid || j <= r && A[j] < A[i]) B[k++] = A[j++] , ans += mid-i+1;
else B[k++] = A[i++];
}
for(i=l;i<=r;i++) A[i] = B[i];
}
int main() {
int A[maxn] , n;
while(~scanf("%d" , &n)) {
for(int i=0;i<n;i++) cin >> A[i];
ans = 0;
merge_sort(A, 0, n-1);
cout << ans << endl;
//for(int i=0;i<n;i++) cout << A[i] << " "; cout << endl;
}
return 0;
}
]]>
]]>
鏈塏縐嶇墿鍝佸拰涓涓閲忎負(fù)V鐨勮儗鍖呫傜i縐嶇墿鍝佹渶澶氭湁n[i]浠跺彲鐢紝姣忎歡璐圭敤鏄痗[i]錛屼環(huán)鍊兼槸w[i]銆傛眰瑙ｅ皢鍝簺鐗╁搧瑁呭叆鑳屽寘鍙嬌榪欎簺鐗╁搧鐨勮垂鐢ㄦ誨拰涓嶈秴榪囪儗鍖呭閲忥紝涓斾環(huán)鍊兼誨拰鏈澶с?/p>
鍩烘湰綆楁硶
榪欓鐩拰瀹屽叏鑳屽寘闂寰堢被浼箋傚熀鏈殑鏂圭▼鍙渶灝嗗畬鍏ㄨ儗鍖呴棶棰樼殑鏂圭▼鐣ュ井涓鏀瑰嵆鍙紝鍥犱負(fù)瀵逛簬絎琲縐嶇墿鍝佹湁n[i]+1縐嶇瓥鐣ワ細(xì)鍙?浠訛紝鍙?浠?#8230;…鍙杗[i]浠躲備護(hù)f[i][v]琛ㄧず鍓峣縐嶇墿鍝佹伆鏀懼叆涓涓閲忎負(fù)v鐨勮儗鍖呯殑鏈澶ф潈鍊鹼紝鍒欐湁鐘舵佽漿縐繪柟紼嬶細(xì)
f[i][v]=max{f[i-1][v-k*c[i]]+k*w[i]|0<=k<=n[i]}
澶嶆潅搴︽槸O(V*Σn[i])銆?/p>
杞寲涓?1鑳屽寘闂
鍙︿竴縐嶅ソ鎯沖ソ鍐欑殑鍩烘湰鏂規(guī)硶鏄漿鍖栦負(fù)01鑳屽寘姹傝В錛氭妸絎琲縐嶇墿鍝佹崲鎴恘[i]浠?1鑳屽寘涓殑鐗╁搧錛屽垯寰楀埌浜嗙墿鍝佹暟涓?#931;n[i]鐨?1鑳屽寘闂錛岀洿鎺ユ眰瑙ｏ紝澶嶆潅搴︿粛鐒舵槸O(V*Σn[i])銆?/p>
浣嗘槸鎴戜滑鏈熸湜灝嗗畠杞寲涓?1鑳屽寘闂涔嬪悗鑳藉鍍忓畬鍏ㄨ儗鍖呬竴鏍烽檷浣庡鏉傚害銆備粛鐒惰冭檻浜岃繘鍒剁殑鎬濇兂錛屾垜浠冭檻鎶婄i縐嶇墿鍝佹崲鎴愯嫢騫蹭歡鐗╁搧錛屼嬌寰楀師闂涓i縐嶇墿鍝佸彲鍙栫殑姣忕絳栫暐——鍙?..n[i]浠?#8212;—鍧囪兘絳変環(huán)浜庡彇鑻ュ共浠朵唬鎹互鍚庣殑鐗╁搧銆傚彟澶栵紝鍙栬秴榪噉[i]浠剁殑絳栫暐蹇呬笉鑳藉嚭鐜般?/p>
鏂規(guī)硶鏄細(xì)灝嗙i縐嶇墿鍝佸垎鎴愯嫢騫蹭歡鐗╁搧錛屽叾涓瘡浠剁墿鍝佹湁涓涓郴鏁幫紝榪欎歡鐗╁搧鐨勮垂鐢ㄥ拰浠峰煎潎鏄師鏉ョ殑璐圭敤鍜屼環(huán)鍊間箻浠ヨ繖涓郴鏁般備嬌榪欎簺緋繪暟鍒嗗埆涓?,2,4,...,2^(k-1),n[i]-2^k+1錛屼笖k鏄弧瓚硁[i]-2^k+1>0鐨勬渶澶ф暣鏁般備緥濡傦紝濡傛灉n[i]涓?3錛屽氨灝嗚繖縐嶇墿鍝佸垎鎴愮郴鏁板垎鍒負(fù)1,2,4,6鐨勫洓浠剁墿鍝併?/p>
鍒嗘垚鐨勮繖鍑犱歡鐗╁搧鐨勭郴鏁板拰涓簄[i]錛岃〃鏄庝笉鍙兘鍙栧浜巒[i]浠剁殑絎琲縐嶇墿鍝併傚彟澶栬繖縐嶆柟娉曚篃鑳戒繚璇佸浜?..n[i]闂寸殑姣忎竴涓暣鏁幫紝鍧囧彲浠ョ敤鑻ュ共涓郴鏁扮殑鍜岃〃紺猴紝榪欎釜璇佹槑鍙互鍒?..2^k-1鍜?^k..n[i]涓ゆ鏉ュ垎鍒璁哄緱鍑猴紝騫朵笉闅撅紝甯屾湜浣犺嚜宸辨濊冨皾璇曚竴涓嬨?/p>
榪欐牱灝卞皢絎琲縐嶇墿鍝佸垎鎴愪簡O(log n[i])縐嶇墿鍝侊紝灝嗗師闂杞寲涓轟簡澶嶆潅搴︿負(fù)<math>O(V*Σlog n[i])鐨?1鑳屽寘闂錛屾槸寰堝ぇ鐨勬敼榪涖?/p>
涓嬮潰緇欏嚭O(log amount)鏃墮棿澶勭悊涓浠跺閲嶈儗鍖呬腑鐗╁搧鐨勮繃紼嬶紝鍏朵腑amount琛ㄧず鐗╁搧鐨勬暟閲忥細(xì)
procedure MultiplePack(cost,weight,amount)
if cost*amount>=V
CompletePack(cost,weight)
return
integer k=1
while k<amount
ZeroOnePack(k*cost,k*weight)
amount=amount-k
k=k*2
ZeroOnePack(amount*cost,amount*weight)
甯屾湜浣犱粩緇嗕綋浼?xì)杩欎釜浼唬鐮佸Q屽鏋滀笉澶悊瑙ｇ殑璇濓紝涓嶅Θ緲昏瘧鎴愮▼搴忎唬鐮佷互鍚庯紝鍗曟鎵ц鍑犳錛屾垨鑰呭ご鑴戝姞綰哥瑪妯℃嫙涓涓嬶紝涔熻灝變細(xì)鎱㈡參鐞嗚В浜嗐?/p>
O(VN)鐨勭畻娉?/h2>
澶氶噸鑳屽寘闂鍚屾牱鏈塐(VN)鐨勭畻娉曘傝繖涓畻娉曞熀浜庡熀鏈畻娉曠殑鐘舵佽漿縐繪柟紼嬶紝浣嗗簲鐢ㄥ崟璋冮槦鍒楃殑鏂規(guī)硶浣挎瘡涓姸鎬佺殑鍊煎彲浠ヤ互鍧囨憡O(1)鐨勬椂闂存眰瑙ｃ傜敱浜庣敤鍗曡皟闃熷垪浼樺寲鐨凞P宸茶秴鍑轟簡NOIP鐨勮寖鍥達(dá)紝鏁呮湰鏂囦笉鍐嶅睍寮璁茶В銆傛垜鏈鍒濅簡瑙ｅ埌榪欎釜鏂規(guī)硶鏄湪妤煎ぉ鎴愮殑“鐢蜂漢鍏”騫葷伅鐗囦笂銆?/p>
灝忕粨
榪欓噷鎴戜滑鐪嬪埌浜嗗皢涓涓畻娉曠殑澶嶆潅搴︾敱O(jiān)(V*Σn[i])鏀硅繘鍒癘(V*Σlog n[i])鐨勮繃紼嬶紝榪樼煡閬撲簡瀛樺湪搴旂敤瓚呭嚭NOIP鑼冨洿鐨勭煡璇嗙殑O(VN)綆楁硶銆傚笇鏈涗綘鐗瑰埆娉ㄦ剰“鎷嗗垎鐗╁搧”鐨勬濇兂鍜屾柟娉曪紝鑷繁璇佹槑涓涓嬪畠鐨勬紜э紝騫跺皢瀹屾暣鐨勭▼搴忎唬鐮佸啓鍑烘潵銆?/p>
鏈塏縐嶇墿鍝佸拰涓涓閲忎負(fù)V鐨勮儗鍖咃紝姣忕鐗╁搧閮芥湁鏃犻檺浠跺彲鐢ㄣ傜i縐嶇墿鍝佺殑璐圭敤鏄痗[i]錛屼環(huán)鍊兼槸w[i]銆傛眰瑙ｅ皢鍝簺鐗╁搧瑁呭叆鑳屽寘鍙嬌榪欎簺鐗╁搧鐨勮垂鐢ㄦ誨拰涓嶈秴榪囪儗鍖呭閲忥紝涓斾環(huán)鍊兼誨拰鏈澶с?/p>
鍩烘湰鎬濊礬
榪欎釜闂闈炲父綾諱技浜?a >01鑳屽寘闂錛屾墍涓嶅悓鐨勬槸姣忕鐗╁搧鏈夋棤闄愪歡銆備篃灝辨槸浠庢瘡縐嶇墿鍝佺殑瑙掑害鑰冭檻錛屼笌瀹冪浉鍏崇殑絳栫暐宸插茍闈炲彇鎴栦笉鍙栦袱縐嶏紝鑰屾槸鏈夊彇0浠躲佸彇1浠躲佸彇2浠?#8230;…絳夊緢澶氱銆傚鏋滀粛鐒舵寜鐓цВ01鑳屽寘鏃剁殑鎬濊礬錛屼護(hù)f[i][v]琛ㄧず鍓峣縐嶇墿鍝佹伆鏀懼叆涓涓閲忎負(fù)v鐨勮儗鍖呯殑鏈澶ф潈鍊箋備粛鐒跺彲浠ユ寜鐓ф瘡縐嶇墿鍝佷笉鍚岀殑絳栫暐鍐欏嚭鐘舵佽漿縐繪柟紼嬶紝鍍忚繖鏍鳳細(xì)
f[i][v]=max{f[i-1][v-k*c[i]]+k*w[i]|0<=k*c[i]<=v}
榪欒窡01鑳屽寘闂涓鏍鋒湁O(VN)涓姸鎬侀渶瑕佹眰瑙ｏ紝浣嗘眰瑙ｆ瘡涓姸鎬佺殑鏃墮棿宸茬粡涓嶆槸甯告暟浜嗭紝姹傝В鐘舵乫[i][v]鐨勬椂闂存槸O(v/c[i])錛屾葷殑澶嶆潅搴﹀彲浠ヨ涓烘槸O(V*Σ(V/c[i]))錛屾槸姣旇緝澶х殑銆?/p>
灝?1鑳屽寘闂鐨勫熀鏈濊礬鍔犱互鏀硅繘錛屽緱鍒頒簡榪欐牱涓涓竻鏅扮殑鏂規(guī)硶銆傝繖璇存槑01鑳屽寘闂鐨勬柟紼嬬殑紜槸寰堥噸瑕侊紝鍙互鎺ㄥ強(qiáng)鍏跺畠綾誨瀷鐨勮儗鍖呴棶棰樸備絾鎴戜滑榪樻槸璇曞浘鏀硅繘榪欎釜澶嶆潅搴︺?/p>
涓涓畝鍗曟湁鏁堢殑浼樺寲
瀹屽叏鑳屽寘闂鏈変竴涓緢綆鍗曟湁鏁堢殑浼樺寲錛屾槸榪欐牱鐨勶細(xì)鑻ヤ袱浠剁墿鍝乮銆乯婊¤凍c[i]<=c[j]涓攚[i]>=w[j]錛屽垯灝嗙墿鍝乯鍘繪帀錛屼笉鐢ㄨ冭檻銆傝繖涓紭鍖栫殑姝ｇ‘鎬ф樉鐒?dòng)灱?xì)浠諱綍鎯呭喌涓嬮兘鍙皢浠峰煎皬璐圭敤楂樺緱j鎹㈡垚鐗╃編浠峰粔鐨刬錛屽緱鍒拌嚦灝戜笉浼?xì)鏇村樊鐨勬栆?guī)銆傚浜庨殢鏈虹敓鎴愮殑鏁版嵁錛岃繖涓柟娉曞線寰浼?xì)澶уぇ鍑彏畱鐗╁搧鐨勪欢鏁板Q屼粠鑰屽姞蹇熷害銆傜劧鑰岃繖涓茍涓嶈兘鏀瑰杽鏈鍧忔儏鍐電殑澶嶆潅搴︼紝鍥犱負(fù)鏈夊彲鑳界壒鍒璁＄殑鏁版嵁鍙互涓浠剁墿鍝佷篃鍘諱笉鎺夈?/p>
榪欎釜浼樺寲鍙互綆鍗曠殑O(N^2)鍦板疄鐜幫紝涓鑸兘鍙互鎵垮彈銆傚彟澶栵紝閽堝鑳屽寘闂鑰岃█錛屾瘮杈冧笉閿欑殑涓縐嶆柟娉曟槸錛氶鍏堝皢璐圭敤澶т簬V鐨勭墿鍝佸幓鎺夛紝鐒跺悗浣跨敤綾諱技璁℃暟鎺掑簭鐨勫仛娉曪紝璁＄畻鍑鴻垂鐢ㄧ浉鍚岀殑鐗╁搧涓環(huán)鍊兼渶楂樼殑鏄摢涓紝鍙互O(V+N)鍦板畬鎴愯繖涓紭鍖栥傝繖涓笉澶噸瑕佺殑榪囩▼灝變笉緇欏嚭浼唬鐮佷簡錛屽笇鏈涗綘鑳界嫭绔嬫濊冨啓鍑轟吉浠ｇ爜鎴栫▼搴忋?/p>
杞寲涓?1鑳屽寘闂姹傝В
鏃㈢劧01鑳屽寘闂鏄渶鍩烘湰鐨勮儗鍖呴棶棰橈紝閭ｄ箞鎴戜滑鍙互鑰冭檻鎶婂畬鍏ㄨ儗鍖呴棶棰樿漿鍖栦負(fù)01鑳屽寘闂鏉ヨВ銆傛渶綆鍗曠殑鎯蟲硶鏄紝鑰冭檻鍒扮i縐嶇墿鍝佹渶澶氶塚/c[i]浠訛紝浜庢槸鍙互鎶婄i縐嶇墿鍝佽漿鍖栦負(fù)V/c[i]浠惰垂鐢ㄥ強(qiáng)浠峰煎潎涓嶅彉鐨勭墿鍝侊紝鐒跺悗姹傝В榪欎釜01鑳屽寘闂銆傝繖鏍峰畬鍏ㄦ病鏈夋敼榪涘熀鏈濊礬鐨勬椂闂村鏉傚害錛屼絾榪欐瘯绔熺粰浜嗘垜浠皢瀹屽叏鑳屽寘闂杞寲涓?1鑳屽寘闂鐨勬濊礬錛氬皢涓縐嶇墿鍝佹媶鎴愬浠剁墿鍝併?/p>
鏇撮珮鏁堢殑杞寲鏂規(guī)硶鏄細(xì)鎶婄i縐嶇墿鍝佹媶鎴愯垂鐢ㄤ負(fù)c[i]*2^k銆佷環(huán)鍊間負(fù)w[i]*2^k鐨勮嫢騫蹭歡鐗╁搧錛屽叾涓璳婊¤凍c[i]*2^k<=V銆傝繖鏄簩榪涘埗鐨勬濇兂錛屽洜涓轟笉綆℃渶浼樼瓥鐣ラ夊嚑浠剁i縐嶇墿鍝侊紝鎬誨彲浠ヨ〃紺烘垚鑻ュ共涓?^k浠剁墿鍝佺殑鍜屻傝繖鏍鋒妸姣忕鐗╁搧鎷嗘垚O(log V/c[i])浠剁墿鍝侊紝鏄竴涓緢澶х殑鏀硅繘銆?/p>
浣嗘垜浠湁鏇翠紭鐨凮(VN)鐨勭畻娉曘?/p>
O(VN)鐨勭畻娉?/h2>
榪欎釜綆楁硶浣跨敤涓緇存暟緇勶紝鍏堢湅浼唬鐮侊細(xì)
for i=1..N
for v=0..V
f[v]=max{f[v],f[v-cost]+weight}
浣犱細(xì)鍙戠幇錛岃繖涓吉浠ｇ爜涓?a >P01鐨勪吉浠ｇ爜鍙湁v鐨勫驚鐜搴忎笉鍚岃屽凡銆備負(fù)浠涔堣繖鏍蜂竴鏀瑰氨鍙鍛紵棣栧厛鎯蟲兂涓轟粈涔圥01涓鎸夌収v=V..0鐨勯嗗簭鏉ュ驚鐜傝繖鏄洜涓鴻淇濊瘉絎琲嬈″驚鐜腑鐨勭姸鎬乫[i][v]鏄敱鐘舵乫[i-1][v-c[i]]閫掓帹鑰屾潵銆傛崲鍙ヨ瘽璇達(dá)紝榪欐鏄負(fù)浜嗕繚璇佹瘡浠剁墿鍝佸彧閫変竴嬈★紝淇濊瘉鍦ㄨ冭檻“閫夊叆絎琲浠剁墿鍝?#8221;榪欎歡絳栫暐鏃訛紝渚濇嵁鐨勬槸涓涓粷鏃犲凡緇忛夊叆絎琲浠剁墿鍝佺殑瀛愮粨鏋渇[i-1][v-c[i]]銆傝岀幇鍦ㄥ畬鍏ㄨ儗鍖呯殑鐗圭偣鎭版槸姣忕鐗╁搧鍙夋棤闄愪歡錛屾墍浠ュ湪鑰冭檻“鍔犻変竴浠剁i縐嶇墿鍝?#8221;榪欑絳栫暐鏃訛紝鍗存闇瑕佷竴涓彲鑳藉凡閫夊叆絎琲縐嶇墿鍝佺殑瀛愮粨鏋渇[i][v-c[i]]錛屾墍浠ュ氨鍙互騫朵笖蹇呴』閲囩敤v=0..V鐨勯『搴忓驚鐜傝繖灝辨槸榪欎釜綆鍗曠殑紼嬪簭涓轟綍鎴愮珛鐨勯亾鐞嗐?/p>
鍊煎緱涓鎻愮殑鏄紝涓婇潰鐨勪吉浠ｇ爜涓袱灞俧or寰幆鐨勬搴忓彲浠ラ鍊掋傝繖涓粨璁烘湁鍙兘浼?xì)甯︽潵绠楁硶鏃堕棿甯告曨C笂鐨勪紭鍖栥?/p>
榪欎釜綆楁硶涔熷彲浠ヤ互鍙﹀鐨勬濊礬寰楀嚭銆備緥濡傦紝灝嗗熀鏈濊礬涓眰瑙[i][v-c[i]]鐨勭姸鎬佽漿縐繪柟紼嬫樉寮忓湴鍐欏嚭鏉ワ紝浠ｅ叆鍘熸柟紼嬩腑錛屼細(xì)鍙戠幇璇ユ柟紼嬪彲浠ョ瓑浠峰湴鍙樺艦鎴愯繖縐嶅艦寮忥細(xì)
f[i][v]=max{f[i-1][v],f[i][v-c[i]]+w[i]}
灝嗚繖涓柟紼嬬敤涓緇存暟緇勫疄鐜幫紝渚垮緱鍒頒簡涓婇潰鐨勪吉浠ｇ爜銆?/p>
鏈鍚庢娊璞″嚭澶勭悊涓浠跺畬鍏ㄨ儗鍖呯被鐗╁搧鐨勮繃紼嬩吉浠ｇ爜錛?/p>
procedure CompletePack(cost,weight)
for v=cost..V
f[v]=max{f[v],f[v-c[i]]+w[i]}
鎬葷粨
瀹屽叏鑳屽寘闂涔熸槸涓涓浉褰撳熀紜鐨勮儗鍖呴棶棰橈紝瀹冩湁涓や釜鐘舵佽漿縐繪柟紼嬶紝鍒嗗埆鍦?#8220;鍩烘湰鎬濊礬”浠ュ強(qiáng)“O(VN)鐨勭畻娉?#8220;鐨勫皬鑺備腑緇欏嚭銆傚笇鏈涗綘鑳藉瀵硅繖涓や釜鐘舵佽漿縐繪柟紼嬮兘浠旂粏鍦頒綋浼?xì)锛屼笉浠呰C綇錛屼篃瑕佸紕鏄庣櫧瀹冧滑鏄庝箞寰楀嚭鏉ョ殑錛屾渶濂借兘澶熻嚜宸辨兂涓縐嶅緱鍒拌繖浜涙柟紼嬬殑鏂規(guī)硶銆備簨瀹炰笂錛屽姣忎竴閬撳姩鎬佽鍒掗鐩兘鎬濊冨叾鏂圭▼鐨勬剰涔変互鍙?qiáng)濡備綍寰楁潵锛屾槸鍔犳繁瀵瑰姩鎬佽鍒掔殑鐞嗚В銆佹彁楂樺姩鎬佽鍒掑姛鍔涚殑濂芥柟娉曘?/p>
鏈塏浠剁墿鍝佸拰涓涓閲忎負(fù)V鐨勮儗鍖呫傜i浠剁墿鍝佺殑璐圭敤鏄痗[i]錛屼環(huán)鍊兼槸w[i]銆傛眰瑙ｅ皢鍝簺鐗╁搧瑁呭叆鑳屽寘鍙嬌浠峰兼誨拰鏈澶с?/p>
鍩烘湰鎬濊礬
榪欐槸鏈鍩虹鐨勮儗鍖呴棶棰橈紝鐗圭偣鏄細(xì)姣忕鐗╁搧浠呮湁涓浠訛紝鍙互閫夋嫨鏀炬垨涓嶆斁銆?/p>
鐢ㄥ瓙闂瀹氫箟鐘舵侊細(xì)鍗砯[i][v]琛ㄧず鍓峣浠剁墿鍝佹伆鏀懼叆涓涓閲忎負(fù)v鐨勮儗鍖呭彲浠ヨ幏寰楃殑鏈澶т環(huán)鍊箋傚垯鍏剁姸鎬佽漿縐繪柟紼嬩究鏄細(xì)
f[i][v]=max{f[i-1][v],f[i-1][v-c[i]]+w[i]}
榪欎釜鏂圭▼闈炲父閲嶈錛屽熀鏈笂鎵鏈夎窡鑳屽寘鐩稿叧鐨勯棶棰樼殑鏂圭▼閮芥槸鐢卞畠琛嶇敓鍑烘潵鐨勩傛墍浠ユ湁蹇呰灝嗗畠璇︾粏瑙ｉ噴涓涓嬶細(xì)“灝嗗墠i浠剁墿鍝佹斁鍏ュ閲忎負(fù)v鐨勮儗鍖呬腑”榪欎釜瀛愰棶棰橈紝鑻ュ彧鑰冭檻絎琲浠剁墿鍝佺殑絳栫暐錛堟斁鎴栦笉鏀撅級錛岄偅涔堝氨鍙互杞寲涓轟竴涓彧鐗墊壇鍓峣-1浠剁墿鍝佺殑闂銆傚鏋滀笉鏀劇i浠剁墿鍝侊紝閭ｄ箞闂灝辮漿鍖栦負(fù)“鍓峣-1浠剁墿鍝佹斁鍏ュ閲忎負(fù)v鐨勮儗鍖呬腑”錛屼環(huán)鍊間負(fù)f[i-1][v]錛涘鏋滄斁絎琲浠剁墿鍝侊紝閭ｄ箞闂灝辮漿鍖栦負(fù)“鍓峣-1浠剁墿鍝佹斁鍏ュ墿涓嬬殑瀹歸噺涓簐-c[i]鐨勮儗鍖呬腑”錛屾鏃惰兘鑾峰緱鐨勬渶澶т環(huán)鍊煎氨鏄痜[i-1][v-c[i]]鍐嶅姞涓婇氳繃鏀懼叆絎琲浠剁墿鍝佽幏寰楃殑浠峰紈[i]銆?/p>
浼樺寲絀洪棿澶嶆潅搴?/h2>
浠ヤ笂鏂規(guī)硶鐨勬椂闂村拰絀洪棿澶嶆潅搴﹀潎涓篛(VN)錛屽叾涓椂闂村鏉傚害搴旇宸茬粡涓嶈兘鍐嶄紭鍖栦簡錛屼絾絀洪棿澶嶆潅搴﹀嵈鍙互浼樺寲鍒癘銆?/p>
鍏堣冭檻涓婇潰璁茬殑鍩烘湰鎬濊礬濡備綍瀹炵幇錛岃偗瀹氭槸鏈変竴涓富寰幆i=1..N錛屾瘡嬈＄畻鍑烘潵浜岀淮鏁扮粍f[i][0..V]鐨勬墍鏈夊箋傞偅涔堬紝濡傛灉鍙敤涓涓暟緇刦[0..V]錛岃兘涓嶈兘淇濊瘉絎琲嬈″驚鐜粨鏉熷悗f[v]涓〃紺虹殑灝辨槸鎴戜滑瀹氫箟鐨勭姸鎬乫[i][v]鍛紵f[i][v]鏄敱f[i-1][v]鍜宖[i-1][v-c[i]]涓や釜瀛愰棶棰橀掓帹鑰屾潵錛岃兘鍚︿繚璇佸湪鎺╢[i][v]鏃訛紙涔熷嵆鍦ㄧi嬈′富寰幆涓帹f[v]鏃訛級鑳藉寰楀埌f[i-1][v]鍜宖[i-1][v-c[i]]鐨勫煎憿錛熶簨瀹炰笂錛岃繖瑕佹眰鍦ㄦ瘡嬈′富寰幆涓垜浠互v=V..0鐨勯『搴忔帹f[v]錛岃繖鏍鋒墠鑳戒繚璇佹帹f[v]鏃秄[v-c[i]]淇濆瓨鐨勬槸鐘舵乫[i-1][v-c[i]]鐨勫箋備吉浠ｇ爜濡備笅錛?/p>
for i=1..N
for v=V..0
f[v]=max{f[v],f[v-c[i]]+w[i]};
鍏朵腑鐨刦[v]=max{f[v],f[v-c[i]]}涓鍙ユ伆灝辯浉褰撲簬鎴戜滑鐨勮漿縐繪柟紼?code>f[i][v]=max{f[i-1][v],f[i-1][v-c[i]]}錛屽洜涓虹幇鍦ㄧ殑f[v-c[i]]灝辯浉褰撲簬鍘熸潵鐨刦[i-1][v-c[i]]銆傚鏋滃皢v鐨勫驚鐜『搴忎粠涓婇潰鐨勯嗗簭鏀規(guī)垚欏哄簭鐨勮瘽錛岄偅涔堝垯鎴愪簡f[i][v]鐢眆[i][v-c[i]]鎺ㄧ煡錛屼笌鏈鎰忎笉絎︼紝浣嗗畠鍗存槸鍙︿竴涓噸瑕佺殑鑳屽寘闂P02鏈綆鎹風(fēng)殑瑙ｅ喅鏂規(guī)錛屾晠瀛︿範(fàn)鍙敤涓緇存暟緇勮В01鑳屽寘闂鏄崄鍒嗗繀瑕佺殑銆?/p>
浜嬪疄涓婏紝浣跨敤涓緇存暟緇勮В01鑳屽寘鐨勭▼搴忓湪鍚庨潰浼?xì)琚娆＄敤鍒板Q屾墍浠ヨ繖閲屾娊璞″嚭涓涓鐞嗕竴浠?1鑳屽寘涓殑鐗╁搧榪囩▼錛屼互鍚庣殑浠ｇ爜涓洿鎺ヨ皟鐢ㄤ笉鍔犺鏄庛?/p>
榪囩▼ZeroOnePack錛岃〃紺哄鐞嗕竴浠?1鑳屽寘涓殑鐗╁搧錛屼袱涓弬鏁癱ost銆亀eight鍒嗗埆琛ㄦ槑榪欎歡鐗╁搧鐨勮垂鐢ㄥ拰浠峰箋?/p>
procedure ZeroOnePack(cost,weight)
for v=V..cost
f[v]=max{f[v],f[v-cost]+weight}
娉ㄦ剰榪欎釜榪囩▼閲岀殑澶勭悊涓庡墠闈㈢粰鍑虹殑浼唬鐮佹湁鎵涓嶅悓銆傚墠闈㈢殑紺轟緥紼嬪簭鍐欐垚v=V..0鏄負(fù)浜嗗湪紼嬪簭涓綋鐜版瘡涓姸鎬侀兘鎸夌収鏂圭▼姹傝В浜嗭紝閬垮厤涓嶅繀瑕佺殑鎬濈淮澶嶆潅搴︺傝岃繖閲屾棦鐒跺凡緇忔娊璞℃垚鐪嬩綔榛戠鐨勮繃紼嬩簡錛屽氨鍙互鍔犲叆浼樺寲銆傝垂鐢ㄤ負(fù)cost鐨勭墿鍝佷笉浼?xì)濯?jiǎng)鍝嶇姸鎬乫[0..cost-1]錛岃繖鏄樉鐒剁殑銆?/p>
鏈変簡榪欎釜榪囩▼浠ュ悗錛?1鑳屽寘闂鐨勪吉浠ｇ爜灝卞彲浠ヨ繖鏍峰啓錛?/p>
for i=1..N
ZeroOnePack(c[i],w[i]);
鍒濆鍖栫殑緇嗚妭闂
鎴戜滑鐪嬪埌鐨勬眰鏈浼樿В鐨勮儗鍖呴棶棰橀鐩腑錛屼簨瀹炰笂鏈変袱縐嶄笉澶浉鍚岀殑闂硶銆傛湁鐨勯鐩姹?#8220;鎭板ソ瑁呮弧鑳屽寘”鏃剁殑鏈浼樿В錛屾湁鐨勯鐩垯騫舵病鏈夎姹傚繀欏繪妸鑳屽寘瑁呮弧銆備竴縐嶅尯鍒繖涓ょ闂硶鐨勫疄鐜版柟娉曟槸鍦ㄥ垵濮嬪寲鐨勬椂鍊欐湁鎵涓嶅悓銆?/p>
濡傛灉鏄涓縐嶉棶娉曪紝瑕佹眰鎭板ソ瑁呮弧鑳屽寘錛岄偅涔堝湪鍒濆鍖栨椂闄や簡f[0]涓?鍏跺畠f[1..V]鍧囪涓?∞錛岃繖鏍峰氨鍙互淇濊瘉鏈緇堝緱鍒扮殑f[N]鏄竴縐嶆伆濂借婊¤儗鍖呯殑鏈浼樿В銆?/p>
濡傛灉騫舵病鏈夎姹傚繀欏繪妸鑳屽寘瑁呮弧錛岃屾槸鍙笇鏈涗環(huán)鏍煎敖閲忓ぇ錛屽垵濮嬪寲鏃跺簲璇ュ皢f[0..V]鍏ㄩ儴璁句負(fù)0銆?/p>
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榪欎釜灝忔妧宸у畬鍏ㄥ彲浠ユ帹騫垮埌鍏跺畠綾誨瀷鐨勮儗鍖呴棶棰橈紝鍚庨潰涔熷氨涓嶅啀瀵硅繘琛岀姸鎬佽漿縐諱箣鍓嶇殑鍒濆鍖栬繘琛岃瑙ｃ?/p>
涓涓父鏁頒紭鍖?/h2>
鍓嶉潰鐨勪吉浠ｇ爜涓湁 for v=V..1錛屽彲浠ュ皢榪欎釜寰幆鐨勪笅闄愯繘琛屾敼榪涖?/p>
鐢變簬鍙渶瑕佹渶鍚巉[v]鐨勫鹼紝鍊掓帹鍓嶄竴涓墿鍝侊紝鍏跺疄鍙鐭ラ亾f[v-w[n]]鍗沖彲銆備互姝ょ被鎺紝瀵逛互絎琷涓儗鍖咃紝鍏跺疄鍙渶瑕佺煡閬撳埌f[v-sum{w[j..n]}]鍗沖彲錛屽嵆浠ｇ爜涓殑
for i=1..N
for v=V..0
鍙互鏀規(guī)垚
for i=1..n
bound=max{V-sum{w[i..n]},c[i]}
for v=V..bound
榪欏浜嶸姣旇緝澶ф椂鏄湁鐢ㄧ殑銆?/p>
灝忕粨
01鑳屽寘闂鏄渶鍩烘湰鐨勮儗鍖呴棶棰橈紝瀹冨寘鍚簡鑳屽寘闂涓璁＄姸鎬併佹柟紼嬬殑鏈鍩烘湰鎬濇兂錛屽彟澶栵紝鍒殑綾誨瀷鐨勮儗鍖呴棶棰樺線寰涔熷彲浠ヨ漿鎹㈡垚01鑳屽寘闂姹傝В銆傛晠涓瀹氳浠旂粏浣撲細(xì)涓婇潰鍩烘湰鎬濊礬鐨勫緱鍑烘柟娉曪紝鐘舵佽漿縐繪柟紼嬬殑鎰忎箟錛屼互鍙?qiáng)鏈鍚庢庢牱浼樺寲鐨勭┖闂村鏉傚害銆?/p>